Multi-level Monte Carlo weak Galerkin method with nested meshes for stochastic Brinkman problem

摘要

This paper is devoted to the numerical analysis of a multi-level Monte Carlo weak Galerkin (MLMCWG) approximation with nested meshes for solving stochastic Brinkman equations with two dimensional spatial domain. With weak gradient operator and a stabilizer at hand, the weak Galerkin (WG) technique is a high-order accurate and stable method which can easily handle deterministic partial differential equations with complex geometries, flows with jump fluid viscosity coefficients or high-contrast permeability fields given by each sample. The multi-level Monte Carlo (MLMC) technique with nested meshes balances the sampling error and the spatial approximation error, where the computational cost can be sharply reduced to log-linear complexity with respect to the degree of freedom in spatial direction. The nested meshes requirement is introduced here in order to simplify the analysis, which can be generalized to MLMC with non-nested meshes. Error estimates are derived in terms of the spatial meshsize and the number of samples. The numerical tests are provided to illustrate the behavior of the MLMCWG method and verify our theoretical results regarding optimal convergence of the approximate solutions.